System and method for efficient non-overlapping partitioning of rectangular regions of interest in multi-channel detection

ABSTRACT

A system and method in a multi-channel detection system for multi-rate filter bank applications for converting overlapping rectangular two-dimensional (2D) regions into a new set of non-overlapping rectangular regions for the efficient reconstruction of a signal wherein each non-overlapping region has a maximum extent in a major dimension is described. Overlapping regions are split into marked regions in a non-uniform grid and merged along the major dimension and along the minor dimension to form non-overlapping regions wherein no two non-overlapping rectangular regions have an adjacent edge orthogonal to the major dimension thereby increasing the efficiency of data compression and reducing error-rates.

BACKGROUND

[0001] In multi-rate filter bank applications (e.g. a widebandchannelizer) where overlapping and non-overlapping 2D rectangularregions represent different frequency bands of interest at differenttimes and over different time durations, different layers of frequencyresolution can potentially generate overlaps causing multiple detectionwithin one time-frequency cell. These time frequency overlaps result inless efficient compression due to multiple transmissions of the samedata. Additionally since signal reconstruction errors increase forsmaller time-frequency regions, the most accurate reconstructioncorresponds to regions with the largest bandwidth and longesttime-duration. FIG. 1 shows twenty exemplary 2D rectangular regions ofinterest showing the overlaps between different regions.

[0002] In binary image coding where the compressed data corresponding tojust the locations and sizes of the non-zero “black” regions issufficient for reconstructing the image, an iterative approach is used.Known methods of binary image coding consists of three main steps: (a) araster-scan through the columns and then the rows of the image to findthe next non-zero pixel corresponding to a top-left corner, (b) acolumn-wise scan to find the top-right corner at the first zero pixeland (c) a row-wise scan to find either the bottom-left or bottom-rightcorner corresponding to a zero pixel between the left and right sides orto a non-zero pixel in the columns directly outside the left and rightsides. However, this method does not provide the set of non-overlappingregions with either maximum vertical-extent or maximumhorizontal-extent. Also, the method cannot be directly applied to a setof overlapping rectangular regions to determine the optimal set ofnon-overlapping regions.

[0003] A known prior art compression technique for binary text imagesuses a similar approach. The prior art technique partitions the non-zeroregions into non-overlapping and fully overlapping regions, defines thevertices and assigns specific codes to the converted rectangularregions' vertices reflective of their status as non-overlapping or fullyoverlapping regions. This method does not provide a set ofnon-overlapping rectangular regions encompassing the entire marked area,nor does it allow for a maximum extent in one dimension.

[0004] For data compression, error reduction, and other reasons, it isdesirable to employ a method for converting overlapping rectangulartwo-dimensional (2D) regions into a new set of non-overlappingrectangular regions to thereby allow for efficient reconstruction of asignal output from the filter bank. It is further desirable that theabove method determine the smallest set of non-overlapping rectangularregions with the maximum extent in either the vertical or the horizontaldimension since signal reconstruction errors are larger for smallertime-frequency regions. The most accurate reconstruction corresponds toregions with the largest bandwidth and longest time duration, i.e.larger time frequency regions.

[0005] Accordingly, it is an object of the disclosed subject matter toobviate many of the above problems in the prior art and to provide anovel method in a multi-channel detection system for transforming aplurality of overlapping two-dimensional rectangular regions intonon-overlapping 2D rectangular regions wherein each non-overlappingregion has a maximum extent in a major dimension (i.e. eitherhorizontally or vertically). An embodiment of the method includes thesteps of: splitting the overlapping regions into marked regions in anon-uniform grid; merging the marked grid regions along the majordimension and along the minor dimension to thereby form non-overlappingregions wherein no two non-overlapping rectangular regions have anadjacent edge orthogonal to the major dimension.

[0006] It is another object of the disclosed subject matter to provide anovel improvement of a method for compressing data. One embodiment ofthe method comprises the step of transforming overlappingtwo-dimensional rectangular regions into non-overlapping 2D rectangularregions wherein the non-overlapping rectangular regions have a maximumextent in one dimension.

[0007] It is yet another object of the disclosed subject matter toprovide, in a time-frequency window of interest, a novel method ofexcising the overlapping portion of two-dimensional rectangular areas.An embodiment of the method comprises the steps of forming a non-uniformtwo-dimensional grid using the coordinates of the overlappingrectangular areas; splitting the overlapping 2D rectangular areas intonon-uniform grid units, and combining adjacent tagged grid units intonon-overlapping rectangular regions defined by major edges and minorcorners.

[0008] It is still another object of the disclosed subject matter toprovide a novel method of reconstructing a coverage area defined byoverlapping two-dimensional rectangular regions with non-overlapping 2Drectangular regions. An embodiment of the method comprises the steps offorming a non-uniform two-dimensional grid using the coordinates of theoverlapping rectangular areas; splitting the overlapping 2D rectangularareas into non-uniform grid units; and combining adjacent tagged gridunits into non-overlapping rectangular regions defined by major edgesand minor corners.

[0009] It is an additional object of the disclosed subject matter toprovide a novel improvement for a method in a Cartesian space defined bya frequency domain and a time domain for transforming a plurality ofoverlapping rectangular regions into a plurality of non-overlappingrectangular regions. An embodiment of the method comprises theimprovement wherein none of the non-overlapping rectangular regionsshare a common edge orthogonal to a preferred dimension.

[0010] It is still an additional object of the disclosed subject matterto provide, in a time-frequency window of interest, a novel method ofexcising the overlapping portion of overlapping two-dimensionalrectangular areas comprising the step of transforming the overlappingrectangular areas into non-overlapping rectangular areas by theimprovement wherein none of the non-overlapping rectangular areas sharea common edge orthogonal to a preferred dimension.

[0011] These and many other objects and advantages of the disclosedsubject matter will be readily apparent to one skilled in the art towhich the disclosure pertains from a perusal or the claims, the appendeddrawings, and the following detailed description of the preferredembodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012]FIG. 1 is a representation of overlapping rectangular regions in anon-uniform grid

[0013]FIG. 2 is a representation of the coverage indicator matrix forthe overlapping rectangular regions in FIG. 1.

[0014]FIG. 3 is a representation of the non-overlapping rectangularregions created from the overlapping rectangular regions in FIG. 1according to an embodiment of the disclosed subject matter.

[0015]FIG. 4 is a representation of non-overlapping rectangular regionscreated from the overlapping rectangular regions in FIG. 1 according toknown prior art.

[0016]FIG. 5 is a graph relating number of non-overlapping rectangularregions obtained via the disclosed subject matter and known prior art.

[0017]FIG. 6 is a graph relating the number of adjacent edges in apreferred dimension for an embodiment of the disclosed subject matterand a known prior art method.

DETAILED DESCRIPTION

[0018] A method according to an embodiment of the disclosed subjectmatter comprises three steps: (1) determining a non-uniform 2D gridcorresponding the all the overlapping rectangular region boundaries, (2)determining the non-uniform grid rectangles covered by one or more ofthe overlapping rectangles and (3) combining directly adjacent coveredgrid regions to find the smallest set of non-overlapping rectangles withthe maximum extent either vertically or horizontally.

[0019] For the i^(th) 2D rectangular region R_(i) 10 in a set of Npossibly overlapping rectangular regions 1 as shown in FIG. 1, letx_(0,i),x_(1,i),y_(0,i),y_(1,i) denote the minimum (left) x-value 11,maximum (right) x-value 12, minimum (bottom) y-value 13 and maximum(top) y-value 14, respectively. The four vectors x₀−[x_(0,1),x_(0,2), .. . ,x_(0,N)]^(T), x₁=[x_(1,1),x_(1,2), . . . ,x_(l,N)]^(T),y₀=[y_(0,1),y_(0,2), . . . , y_(1,N)]^(T) represent the 4N cornerlocations of all N rectangles (where [ ]^(T) is the transpose vector).It is desirable to find the set of non-overlapping rectangular regionscovering the same areas as the overlapping regions and with the maximumextent in the “major” dimension. Without loss of generality, thevertical dimension is assumed to be the “major” dimension in the currentdiscussion, wherein the vertical dimension or horizontal dimension maycorrespond to parameters such as time or frequency. The other dimensionis then defined as the minor dimension. For the case where the “major”dimension is horizontal, x₀ is interchanged with y₀ and x₁ with y₁.

[0020] Determining a Non-uniform Grid

[0021] An initial step in an embodiment of the disclosure is to define anon-uniform grid corresponding to the unique x-values and uniquey-values of a set of possibly overlapping rectangular regions. Let theN_(x)×1 vector x_(g) denote the unique x values in the 2N×1 vector [x₀^(T),x₁ ^(T)]^(T), sorted in ascending order, i.e.min(x₀)=x_(g,1)<x_(g,2)<. . . <x_(g,N) _(x) =max(x₁). Similarly, let theN_(y)×1 vector y_(g) denote the sorted unique y values in the vector [y₀^(T),y₁ ^(T)]^(T), i.e. min(y₀)=y_(g,1)<y_(g,2)<. . . <y_(g,N) _(y)=max(y₁). FIG. 1 shows an example of a set of 20 randomly generatedoverlapping rectangular regions (i.e. N=20), some of which overlapothers, and the resulting non-uniform 2D grid derived from the unique xand y values of the rectangular regions. The non-uniform grid is shownby the dashed-lines 5 and tick-marks along each axis. The solid linesindicate the edges of the different rectangular regions. By thedefinitions of x_(g) and y_(g), the number of grid points is(N_(x)N_(y))≦(2N)². In FIG. 1, N_(x)=N_(y)=29 and 2N=40.

[0022] Determining the Non-uniform Grid Regions Covered by Rectangles

[0023] A latter step in the process is to determine which singlerow/single column non-uniform grid regions are covered by one or more ofthe rectangular regions represented by x₀, x₁, y₀, and y₁. Let theN_(y)×N_(x) matrix C be a coverage indicator matrix where C_(i,j)=1 ifx_(0,n)≦x_(g,j)<x_(1,n) and y_(0,n)≦y_(g,i)<y_(1,n) for any n−1, . . . ,N and otherwise C_(i,j)=0. The small dot symbols 22 in FIG. 2 indicatethe non-zero elements in C 20 for the rectangular regions shown in FIG.1, note that the x and y axes of FIG. 2 have been transformed torepresent the cardinality of the unique x and y values, respectively.The last row and last column of C consist of all zeros (shown as blanks)in FIG. 2, since x_(0,n)<x_(1,n)≦x_(g,N) _(x) andy_(0,n)<y_(1,n)≦y_(g,N) _(y) for any n=1, . . . , N. At this point, aset of non-overlapping single column or single row rectangles can befound directly from the row and column indices of the non-zero elementsof matrix C 20. However, this set of covered grid rectangle regionscorresponds to the largest partitioning of the covered regions. A muchmore efficient partitioning results from grouping multiple covered gridregions that are directly adjacent to each other.

[0024] Combining Directly Adjacent Covered Grid Regions

[0025] To determine a smaller set of non-overlapping rectangles,adjacent covered grid regions are grouped or merged, first in the major(vertical) dimension and second in the minor (horizontal) dimension.Again the major and minor dimensions are assigned for illustration onlyand are not intended to be limiting the embodiment of the disclosedsubject matter in anyway. Grouping adjacent covered grid regions can beequivalently expressed in terms of edge-detection for the “binary image”formed by the coverage matrix C 20. The top and bottom “edges” in C 20correspond to the non-zero 1^(st)-order differences in the rows of C 20.Since any “ones” (small dots 22) in the 1^(st) row of C 20 correspond tobottom edges of tall-narrow single column rectangles, let the1^(st)-order row-difference matrix be defined as$\lbrack C_{\Delta \quad y} \rbrack_{i,j} = \{ \begin{matrix}{C_{i,j},} & {i = 1} \\{{C_{i,j} - C_{{i - 1},j}},} & {i > 1}\end{matrix} $

[0026] The rows of the non-zero elements of C_(Δy) 30 correspond toeither bottom edges 31, where [C_(Δy)]_(i,j)=1, or top edges 32, where[C_(Δy)]_(i,j)=−1, as shown in FIG. 2. The set ofmultiple-row/single-column non-overlapping rectangles can be representedby the row and columns indices of the non-zero elements of C_(Δy) 30.Let the N_(C)×1 vectors e_(y) ₀ and e_(y) ₁ denote the row indicescorresponding to positive 1 and negative 1 elements in C_(Δy) 30, i.e.the bottom edges and top edges, respectively. Let the N_(C)×1 vectore_(x) ₀ likewise denote the column indices corresponding to the positive“1” elements in C_(Δy) 30. It is assumed that the index vector e_(x) ₀is formed via a “raster-scan” down the 1^(st) column of C_(Δy) 30, thenthe 2^(nd) column, and so on.

[0027] The next step is to group any multiple-row/single-columnrectangles in adjacent columns that have identical row indices. This canbe performed via a corner-detection process similar to the previousedge-detection step. Since the corners of themultiple-row/multiple-column rectangles are desired, 1^(st)-orderdifferences are computed across the columns of C_(Δy) rather than Citself. Let the N_(y)×N_(x) matrix C_(ΔxΔy) 40 denote the column-wise1^(st)-order differences of C_(Δy) 30, i.e.$\lbrack C_{\Delta \quad x\quad \Delta \quad y} \rbrack_{i,j} = \{ \begin{matrix}{\lbrack C_{\Delta \quad y} \rbrack_{i,j},} & {j = 1} \\{{\lbrack C_{\Delta \quad y} \rbrack_{i,j} - \lbrack C_{\Delta \quad y} \rbrack_{i,{j - 1}}},} & {j > 1}\end{matrix} $

[0028] The bottom-left and top-right corners correspond to whereC_(ΔxΔy)=1 while the top-left and bottom-right corners correspond towhere C_(ΔxΔy)=−1. The locations of the corners, as well as the top andbottom edges, are shown in FIG. 2. The squares, triangles and circlescorrespond to non-zero elements of C 20, C_(Δy) 30, and C_(ΔxΔy) 40,respectively. The top and bottom edges are further indicated by theorientation of the triangle symbols. The bottom edges coincide withnon-zero elements of C 20, while the top-edges do not.

[0029] Given the matrix C_(ΔxΔy) 40 and index vectors e_(y) ₀ , e_(y) ₁, and e_(x) ₀ , the multiple-row/multiple-column non-overlappingrectangles can be determined according to an embodiment of the disclosedsubject matter via the following procedure.

[0030] Let n=1 and let N_(y)×N_(x) matrix D=0.

[0031] For i=1, . . . N_(C), let i_(y)=[e_(y) ₀ ]_(i) and i_(x)=[e_(x) ₀]_(i)

[0032] If D_(i) _(y) _(,i) _(x) =0, then

[0033] Assign [{tilde over (e)}_(y) ₀ ]_(n)=[e_(y) ₀ ]_(i), [{tilde over(e)}_(y) ₁ ]_(n)=[e_(y) ₁${{Let}\quad b_{j}} = \{ {\begin{matrix}{0,} & {j \leq i_{x}} \\{{\sum\limits_{m = i_{y}}^{m_{y}}{\lbrack C_{\Delta \quad x\quad \Delta \quad y} \rbrack_{m,j}}},} & {j > i_{x}}\end{matrix}.} $

[0034] Let m_(x) denote the index of the first non-zero element ofvector b.

[0035] Assign [{tilde over (e)}_(x) ₁ ]_(n)=m_(x) and D_(r,c)=1 fori_(y)≦r≦m_(y) and i_(x)≦c≦m_(x).

[0036] Increment n=n+1

[0037] Each element of matrix D indicates if the grid-regioncorresponding to that row and column has already been assigned to amultiple-row/multiple-column rectangle. The N_(x)×1 vector b indicatesif matrix C_(ΔxΔy) 40 has any non-zero elements from row i_(y) to rowm_(y) in the columns greater than i_(x). It is used to find theright-edge of the multiple-row/multiple-column rectangle withbottom-left at (i_(x),i_(y)) and top-left at (i_(x),m_(y)). The vectors,{tilde over (e)}_(x) ₀ , {tilde over (e)}_(y) ₀ , {tilde over (e)}_(x) ₁, and {tilde over (e)}_(y) ₁ consist of the indices corresponding to thebottom-left and top-right corners of the multiple-row/multiple-columnnon-overlapping rectangles. The non-overlapping rectangles on thenon-uniform grid correspond to the vectors {tilde over (x)}₀, {tildeover (x)}₀, {tilde over (y)}₀ and {tilde over (y)}₁ with elements givenby [{tilde over (x)}₀]_(i)=x_(g)([{tilde over (e)}_(x) ₀ ]_(i)),[{tildeover (x)}₁]_(i)=x_(g)([{tilde over (e)}_(x) ₁ ]_(i)), [{tilde over(y)}₀]_(i)=y_(g)([{tilde over (e)}_(y) ₀ ]_(i)) and [{tilde over(y)}₁]_(i)=y_(g)([{tilde over (e)}_(y) ₁ ]_(i)), respectively.

[0038] For comparison, non-overlapping regions determined based on theprior art approach are shown in FIG. 4. The prior art and the disclosedmethods differ in how the non-overlapping regions are determined fromthe coverage indicator matrix C.

[0039] For the example of rectangles regions shown in FIG. 1, theresulting non-overlapping rectangles, computed via the steps above foran embodiment of the disclosed subject matter, are shown in FIG. 3. Allthe non-overlapping rectangles in FIG. 3 extend over multiple verticalgrid regions and several extend over multiple horizontal grid regions.

[0040] The rectangles in FIG. 3 may be directly adjacent to each otherin the minor dimension, i.e. horizontally (with a right-edge against aleft-edge), but not in the major dimension, i.e. vertically (with atop-edge against a bottom-edge). In other words, between any tworectangles there are no adjacent edges orthogonal to the majordimension. This feature is desirable in signal reconstruction frommulti-rate filter banks where the errors in the reconstructed signaltend to increase with channelization into narrower bandwidth channels.Similarly with reconstruction filter banks, the reconstruction improvesfor longer time durations so the desired rectangular regions should havethe maximum time-extent for each sub-channel.

[0041] The non-overlapping regions determined from matrix C using theprior art method are shown in FIG. 4. While the number ofnon-overlapping rectangles is smaller with the prior art approach, 23versus 25 for the above described embodiment of the inventive method,the vertical extent (major dimension) is not maximized for a fixed valueof x using the prior art approach. This can be seen from the occurrenceof regions 45, as shown in FIG. 4, that are adjacent to other regionsdirectly above or below, i.e. with top-edges against bottom edges.

[0042] An embodiment of the disclosed subject matter generally gives alarger number of rectangles due to the constraint on the extent of therectangles in the major dimension. This can be seen from the twohistograms shown in FIG. 5, where the mode of the embodiment of thedisclosed subject matter's histogram is generally to the right of themode of the prior art histogram. However, the number of non-overlappingregions resulting from using the above described inventive embodiment ofthe disclosed subject matter is typically only slightly larger than thenumber of non-overlapping regions resulting from using the prior art.

[0043] The performance of the two methods with respect to maximizing theextent of the non-overlapping regions in the major dimension can bemeasured from the number of undesirable shared edges between any tworegions. When the major dimension is vertical, this corresponds to thenumber of times a non-overlapping region is directly above or belowanother region, i.e. bottom-edge against top-edge. In FIG. 6, thehistograms of the number of undesirable shared edges are shown for theprior art and an inventive embodiment of the disclosed subject matter.Two histograms are shown for the prior art, corresponding torow-then-column and column-then-row raster-scans. Based on 1000 MonteCarlo trials, the method according to an embodiment of the disclosedsubject matter had no undesirable edges, adjacent edges orthogonal tothe major dimension. In contrast, the prior art results in adjacentnon-overlapping regions in the major dimension regardless of the orderof the raster-scan.

[0044] In an embodiment of the disclosed subject matter, rectangularregions defining bandwidth, time slots or other particular sets ofvalues, may likewise by implemented. Hard indices can be established forrectangular regions which restrict merging with adjacent covered regionsin the dimension of interest. An embodiment can also use erosion and/ordilation morphological operations on the coverage indicator matrix, or“image”, to avoid situations with many closely spaced but not directlyadjacent time-frequency regions corresponding to greater computationthan that for a few larger time-frequency regions over the same areas.

[0045] In another embodiment of the disclosed subject matter, the abovedescribed procedure may be implemented in machine readable softwarecode, in firmware, or in hardware including, but not limited tointegrated circuits (IC), application specific integrated circuits(ASICs), printed wiring boards (PWB), discrete logic circuits, etc.

[0046] While preferred embodiments of the present invention have beendescribed, it is to be understood that the embodiments described areillustrative only and that the scope of the invention is to be definedsolely by the appended claims when accorded a full range of equivalence,many variations and modifications naturally occurring to those of skillin the art from a perusal thereof.

We claim:
 1. In a multi-channel detection system, a method oftransforming a plurality of overlapping two-dimensional rectangularregions positioned in a plane with two orthogonal axes intonon-overlapping two-dimensional rectangular regions wherein eachnon-overlapping region has a maximum extent in a major dimensionoriented parallel to one of the orthogonal axes comprising the steps of:(a) splitting the overlapping regions into marked regions in anon-uniform grid; (b) merging the marked regions along the majordimension and along the minor dimension to form non-overlappingrectangular regions, wherein no two non-overlapping rectangular regionshave an adjacent edge orthogonal to the major dimension.
 2. The methodof claim 1 wherein the step of creating the marked regions comprises thesteps of: (a) obtaining for each of the overlapping rectangular regions,corner coordinates comprising a major dimension component and a minordimension component; (b) transforming the coordinates into respectivenon-uniform grid indices by assigning each major dimension componentvalue a corresponding major dimension component value in the non-uniformgrid and assigning each minor dimension component value a correspondingminor dimension component value in the non-uniform grid; (c) markingrectangular area defined by the corresponding non-uniform grid indicesfor each of the plurality of overlapping rectangular regions.
 3. Themethod of claim 1 wherein the step of merging the marked regions alongthe major dimension comprises the steps of: (a) establishing sequentialminor non-uniform grid indices along the axis orthogonal to the majordimension corresponding cardinally to the unique minor axis coordinatesof the overlapping rectangular regions and establishing sequential majornon-uniform grid indices along the major dimension correspondingcardinally to the unique major axis coordinates of the overlappingrectangular regions; (b) joining adjacent marked regions sharing thesame minor axis non-uniform grid indices along the major axis to createsets of marked regions along the major dimension, each set bound in themajor dimension by the greatest and least major non-uniform grid indicesof the marked regions within the respective set; (c) joining sets ofmarked regions with the same major dimension bounds along the minordimension to create non-overlapping rectangular regions; (d) obtainingthe corner coordinates for each of the non-overlapping rectangularregions.
 4. The method of claim 2, wherein the corresponding majordimension non-uniform grid indices are sequential integers and thecorresponding minor dimension non-uniform grid indices are sequentialintegers.
 5. The method of claim 1, wherein the major dimension isvertical and the minor is horizontal.
 6. The method of claim 1, whereinone of the orthogonal axis represents time and the other orthogonal axisrepresents frequency.
 7. A method of compressing data comprising thestep of transforming a set of overlapping two dimensional rectangularregions positioned in a plane defined by two orthogonal axes into a setof non-overlapping two dimensional rectangular regions, the improvementwherein the non-overlapping rectangular regions comprise a maximumextent in one dimension parallel to one of the orthogonal axes.
 8. In atime-frequency window of interest, a method of excising overlappingportions of a set of two dimensional rectangular areas positioned in aplane with two orthogonal axes defining a major dimension and a minordimension comprising the steps of: (a) forming a non-uniform twodimensional grid using corner coordinates defining each of therectangular areas in the set; (b) transforming a coverage area of theset into a corresponding coverage area in the non-uniform grid, therebycreating a plurality of covered non-uniform grid units; (c) combiningadjacent covered non-uniform grid units into a second set ofnon-overlapping rectangular regions defined by edges in the majordimension and corners in the minor dimension.
 9. The method of claim 8,wherein the major dimension is a frequency domain and the minordimension is a time domain.
 10. The method of claim 8, wherein the majordimension is a time domain and the minor dimension is a frequencydomain.
 11. The method of claim 8, wherein no two of the rectangularareas in the set of the non-overlapping rectangular areas share anadjacent edge orthogonal to the major dimension.
 12. The method of claim8, wherein the step of forming a non-uniform two dimensional gridcomprises the steps of: (a) ordering distinct major dimension cornercoordinate values defining the rectangular areas in the set ofoverlapping rectangular areas and assigning sequential integers to eachof the distinct major dimension coordinate values; (b) ordering distinctminor dimension corner coordinate values defining the rectangular areasin the set of overlapping rectangular areas and assigning sequentialintegers to each of the distinct minor dimension coordinate values. 13.The method of claim 12, wherein the step of transforming a coverage areaof the set into a corresponding coverage area in the non-uniform gridincludes the steps of: (a) replacing the major and minor dimensioncorner coordinate values defining the rectangular areas with theassigned integers, thereby creating non-uniform rectangular areacoordinates; and, (b) covering each non-uniform grid unit within thenon-uniform rectangular area defined by the non-uniform rectangular areacoordinates, thereby forming covered non-uniform grid units.
 14. Themethod of claim 8, wherein the step of combining adjacent coverednon-uniform grid units into a second set of non-overlapping rectangularregions defined by edges in the major dimension and corners in the minordimension comprises the steps of: (a) combining adjacent coverednon-uniform grid units along the major dimension there by formingcovered sets of covered non-uniform rectangular area, each set onenon-uniform grid unit in width in the minor dimension bound in by majordimension edges in the major dimension; (b) combining the adjacentcovered sets strips sharing the same major edges in the minor dimensionforming a second set of non-overlapping rectangular regions with amaximum extent in the major dimension.
 15. A method of reconstructing acoverage area of a set of two dimensional overlapping rectangularregions on a plane with two orthogonal axes defining a major dimensionand a minor dimension with a set of non-overlapping rectangular regionscomprising the steps of: define non-uniform grid on the orthogonal axes;define the coverage area on the non-uniform grid; create singlecolumn/single row rectangles matching the coverage area on thenon-uniform grid; assign coordinates to the corners of the singlecolumn/single row rectangles; for each single column, combine contiguoussingle column/single row rectangles to thereby form at least one multirow/single column rectangle; combine contiguous multi-row/single columnrectangles that have the same major dimension coordinate to form the setof non-overlapping rectangular regions.
 16. The method of claim 15,wherein the rows are along the minor axis and column are along the majordimension.
 17. In a Cartesian space defined by a frequency domain and atime domain, a method of transforming a plurality of over-lappingrectangular regions, in a plane with two orthogonal axes, one of theaxes parallel to a preferred dimension, into a plurality ofnon-overlapping rectangular regions, the improvement wherein none of thenon-overlapping rectangular regions share a common edge orthogonal tothe preferred dimension.
 18. The method of claim 17, wherein theorthogonal axes are a time domain and a frequency domain and thepreferred dimension is parallel to the time domain.
 19. The method ofclaim 17, wherein the orthogonal axes are a time domain and a frequencydomain and the preferred dimension is parallel to the frequency domain.20. The method of claim 17, comprising the step of superimposing avirtual grid onto the Cartesian space, wherein each unit of the virtualgrid corresponds to an edge of one or more of the plurality ofoverlapping rectangular regions.
 21. The method of claim 20, comprisingthe step of joining adjacent virtual grid units in the preferreddimension that are covered by the plurality of overlapping rectangularregions forming a plurality virtual strips.
 22. The method of claim 21,comprising the step of joining adjacent virtual strips in anotherdimension that share the same coverage in the preferred dimensionthereby forming virtual non-overlapping rectangular regions in the grid.23. The method of claim 22, comprising the steps of determiningcorresponding coordinates in the Cartesian space for the virtualnon-overlapping rectangular regions.
 24. In a time-frequency window ofinterest, a method of excising the overlapping portion of twodimensional rectangular areas within a plane defined by orthogonal axesof time and frequency, and a preferred dimension parallel to one of theorthogonal axes, comprising the step of transforming the rectangularareas into non-overlapping rectangular areas, the improvement whereinnone of the non-overlapping rectangular areas share a common edgeorthogonal to a preferred dimension.
 25. A method of transforming aplurality of rectangular regions, wherein two or more of the rectangularregions overlap, into a plurality of non-overlapping rectangular regionscomprising the steps of: determining a non-uniform two dimensional gridcorresponding the rectangular region boundaries; determining non-uniformgrid rectangles covered by one or more of the plurality of rectangularregions; and, combining directly adjacent covered grid rectangles toobtain the smallest set of non-overlapping rectangles with a maximumextent in a major dimension.
 26. The method of claim 25 comprising thestep of determining a coverage indicator matrix C.
 27. The method ofclaim 26 comprising the step of using C to determine an first orderdifference matrix C_(Δy) defining the extent in the major dimension ofthe non-overlapping rectangular regions.
 28. The method of claim 27,comprising the step of determining a second first order differencematrix C_(ΔxΔy) defining the corners.
 29. The method of claim 28,comprising the step of using C, C_(Δy), C_(ΔxΔy) to determine indicesfor regions with adjacent covered grids in the major dimension.
 30. Themethod of claim 29, wherein the regions with adjacent covered grids arecombined with adjacent regions with the same indices.
 31. Amulti-channel detection system for transforming a plurality ofoverlapping two-dimensional rectangular regions into non-overlapping twodimensional rectangular regions wherein each non-overlapping region hasa maximum extent in a major dimension parallel to one of two orthogonalaxes, comprising: means for splitting the overlapping regions intomarked regions in a non-uniform grid; and, means for merging the markedgrid regions along the major dimension and along the minor dimension toform non-overlapping regions, wherein no two of the non-overlappingrectangular region have adjacent edges orthogonal to the majordimension.
 32. A time-frequency window of interest for excising theoverlapping portion of two-dimensional rectangular areas comprising:means for forming a non-uniform two-dimensional grid using the cornercoordinates of the overlapping rectangular areas; means for splittingthe overlapping 2D rectangular areas into covered non-uniform gridunits; and, means for combining adjacent covered non-uniform grid unitsinto non-overlapping rectangular regions defined by major edges andminor corners.
 33. A system for reconstructing a coverage area definedby overlapping two dimensional rectangular regions with non-overlapping2-D rectangular regions comprising: means for forming a non-uniform 2dimensional grid using the coordinates of the overlapping rectangularareas; means for splitting the overlapping 2D rectangular areas intonon-uniform grid units; and means for combining adjacent tagged gridunits into non-overlapping rectangular regions defined by major edgesand minor corners.
 34. A system for transforming a plurality ofrectangular regions, wherein two or more of the rectangular regionsoverlap, into a plurality of non-overlapping rectangular regionscomprising: means for determining a non-uniform 2D grid correspondingthe rectangular region boundaries; means for determining non-uniformgrid rectangles covered by one or more of the plurality of rectangularregions; and, means for combining directly adjacent covered gridrectangles to obtain the smallest set of non-overlapping rectangles witha maximum extent in a major dimension.